A repunit is a whole number consisting of only 1's, such as 1, 111, or 111111111. (It's like a repeating unit.)
These numbers have some fun properties. For instance, many have already noted that the square of a repunit exhibits a nice pattern:
111*111 = 12321,
1111*1111 = 1234321,
11111*11111 = 123454321,
though this pattern is somewhat messed up by carrying digits when the repunit size is bigger than 9.
Here's another less well known pattern:
222+(333)^{2} = 111111,
2222+(3333)^{2} = 11111111,
22222+(33333)^{2} = 1111111111.
You might see if you can figure out why.
Presentation Suggestions:
Can you find other cool properties of repunits?
The Math Behind the Fact:
This property and many such properties can be proved by writing an ndigit repunit as:
(10^{n}  1 )/9
which follows from noting the repunit is a sum of a geometric series with ratio 10. For n=4, this is just the identity 1111 = 9999/9.
You might also check out the Fun Facts
Multiplication by 11 and
Multiplication by 111.
How to Cite this Page:
Su, Francis E., et al. "Repunit Fun."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
